Riemann problem with a degenerate coefficient on a Riemann surface (Q1357919)

Let \(R\) be the Riemann surface of a genus \(h\leq \infty\), \(\Gamma\) be a contour on \(R\), consisting of a finite number of smooth closed components with certain orientation. Suppose that on \(\Gamma\) the holomorphic functions \(\rho_+\), \(\rho_-\) and \(\rho_0\) are given, having finitely many zeros \(\Gamma\). The author considers, in a weighted Hölder space the following problem: Find a function \(\Phi\) meromorphic on \(\mathbb{R} \backslash \Gamma\) such that the limit values on \(\Gamma\) satisfy the condition \(a\Phi^+ -b\Phi^-=g\), where \(a= \rho_+ \rho_0a_0\), \(b= \rho_- \rho_0b_0\); \(a_0,b_0,g\) are given functions. He obtains a Noetherian theorem and gives a formula for the calculation of the index of the problem.